Transcript Course
MTH 251 – Differential Calculus
Chapter 4 – Applications of Derivatives Section 4.4
Concavity and Curve Sketching
Copyright © 2010 by Ron Wallace, all rights reserved.
Interpretation of Derivatives …
• 1 st Derivative:
0 or DNE
critical point (potential local extremum) Positive Negative
increasing decreasing
• 2 nd Derivative?
Describes the behavior of the 1 st Positive
1 st derivative derivative is increasing
• function may be increasing at a faster rate • function may be decreasing at a lower rate
Negative
1 st derivative is decreasing
• function may be increasing at a slower rate • function may be decreasing at a faster rate
This characteristic is called:
concavity
Concave Up
A behavior of a function’s curvature.
• The derivative of
f(x)
is increasing.
If f ’(x) > 0 , the function is getting bigger faster.
If f ’(x) < 0 , the function is getting smaller slower.
• Second derivative is positive.
f ’’(x) > 0
Tangents are “below” the function.
The function is curving in a counter-clockwise direction.
It has the potential to “hold water”.
Concave Down
A behavior of a function’s curvature.
• The derivative of
f(x)
is decreasing.
If f ’(x) > 0 , the function is getting bigger slower.
If f ’(x) < 0 , the function is getting smaller faster.
• Second derivative is negative.
f ’’(x) < 0
Tangents are “above” the function.
The function is curving in a clockwise direction.
It will “spill water”.
Inflection Points
• A point where the concavity changes.
from up to down or from down to up
• Where can this occur? (wrt derivatives)
f ’’(x) = 0 f ’’(x) DNE
• Finding inflection points …
1. Determine
f ’’(x)
2. Solve
f ’’(x) = 0
3. Check values of
f ’’(x)
on each side of each solution of step 2.
• Example … find the inflection points of … 1 5
x
5 3 4
x
4 2 3
x
3
Concavity & Local Extrema
• Concavity at a local maximum?
Down … f ’’(x) < 0
• Concavity at a local minimum?
Up … f ’’(x) > 0
The Second Derivative Test
f ‘(c) = 0 & f ’’(c) < 0
f ‘(c) = 0 & f ’’(c) > 0
f ‘(c) = 0 & f ’’(c) = 0
local maximum at x = c local minimum at x = c test fails (i.e. no conclusion)
10 2
Using the 2
nd
Derivative Test to Determine Local Extrema
• Method:
1. Determine
f ’(x)
&
f ’’(x)
2. Solve
f ’(x) = 0
3. Check the values of
f ’’(x)
at the solutions from step 2.
• Example - Use the second derivative test to find the local extrema of the following function.
4
x
3
x
4
Curvature
Actually, covered in chapter 13 in multi-dimensions. This is the 2-D equivalent.
• Measurement of the “bentness” of a curve.
1
d
2
y dx
2
dy dx
2 3 2 Radius of Curvature 1
One More Example
• Completely describe the behavior of the function …
x f
(
x
)
xe
Critical points Increasing/Decreasing Concavity Inflection Points Local & Global Extrema Curvature